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MathGroup Archive 2007

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Re: Mind+Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78085] Re: [mg78010] Mind+Mathematica
  • From: DrMajorBob <drmajorbob at bigfoot.com>
  • Date: Fri, 22 Jun 2007 06:48:03 -0400 (EDT)
  • References: <28708894.1182424059011.JavaMail.root@m35>
  • Reply-to: drmajorbob at bigfoot.com

Or (in v6):

step1 = Sin[z]*Sin[z^3 + z] // TrigExpand // Expand

Cos[z^3]/2 - 1/2 Cos[z]^2 Cos[z^3] + 1/2 Cos[z^3] Sin[z]^2 +
  Cos[z] Sin[z] Sin[z^3]

step2 = Integrate[#, {z, 0, Infinity}] & /@ step1

(\[Pi] (AiryAi[-2/3^(1/3)] - AiryAi[2/3^(1/3)]))/(4 3^(1/3)) +
  Gamma[1/3]/(
  4 Sqrt[3]) + (-3^(
     1/6) \[Pi] (AiryAi[-2/3^(1/3)] + AiryAi[2/3^(1/3)]) +
   Gamma[1/3])/(
  8 Sqrt[3]) + (\[Pi] (-6 +
     3^(2/3) (AiryAi[-2/3^(1/3)] + AiryAi[2/3^(1/3)]) Gamma[-1/3]))/(
  72 Gamma[2/3])

step3 = step2 // FullSimplify

1/6 \[Pi] (-3^(2/3) AiryAi[2/3^(1/3)] - 3/Gamma[-1/3])

step3 // N

0.295741

Bobby

On Thu, 21 Jun 2007 04:45:59 -0500, dimitris <dimmechan at yahoo.com> wrote:

> The integral
>
> Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}]
>
> (as I was informed)
>
> gives a incorrectly divergent message.
> The integral however is convergent.
>
> The following is part of my response to another forum.
> Demonstrate how vital is to help Mathematica sometimes.
>
> In[2]:=
> $Version
>
> Out[2]=
> "5.2 for Microsoft Windows (June 20, 2005)"
>
> In[3]:=
> int=Integrate[Sin[z]*Sin[z^3 + z], {z, 0, Infinity}](*the integral
> stays unevaluated*)
>
> Out[3]=
> Integrate[Sin[z]*Sin[z + z^3], {z, 0, Infinity}]
>
> In[3]:=
> int2 = (int /. Integrate[f_, x_] :> Integrate[#1, {z, 0, Infinity}]
> & ) /@ Expand[Sin[z]*TrigExpand[Sin[z^3 + z]]]
>
> Out[3]=
> (1/72)*(2*Sqrt[6]*Pi*(BesselI[1/3, (4*Sqrt[2/3])/3] - BesselJ[1/3,
> (4*Sqrt[2/3])/3]) +
>     3*Gamma[1/3]*(2*Sqrt[3] - Sqrt[2]*BesselI[-(1/3), (4*Sqrt[2/3])/
> 3]*Gamma[2/3] - Sqrt[2]*BesselJ[-(1/3), (4*Sqrt[2/3])/3]*Gamma[2/3]))
> + Integrate[Cos[z]*Sin[z]*Sin[z^3], {z, 0, Infinity}]
>
> In[4]:=
> int3 = (1/2)*Integrate[Sin[2*z]*Sin[z^3], {z, 0, Infinity}]
>
> Out[4]=
> (Pi*(AiryAi[-(2/3^(1/3))] - AiryAi[2/3^(1/3)]))/(4*3^(1/3))
>
> In[5]:=
> FullSimplify[int2 /. Integrate[x___] :> int3]
>
> Out[5]=
> (-2*3^(1/6)*Pi*AiryAi[2/3^(1/3)] + Gamma[1/3])/(4*Sqrt[3])
>
> In[6]:=
> N[%, 40]
>
> Out[6]=
> 0.295741225849781931593673891336119670357883693300484102195`40.
>
> Brought to you by M^2
> (Man+Mathematica!)
>
> Dimitris
>
> PS
> I spent almost two hours to figure out a workaround.
> How ancient Greeks said:
> "It is not easy to get Goods"
>
> PS2
> Enjoy Mathematics and Mathematica!
>
>
>



-- =

DrMajorBob at bigfoot.com


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